Question

1. A box without a top is to be made from a rectangular piece of cardboard, with dimensions 8 in. by 10 in., by cutting out square corners with side length x and folding up the sides.

(a) Write an equation for the volume V of the box in terms of x.
(b) Use technology to estimate the value of x, to the nearest tenth, that gives the greatest volume. Explain your process.

Answer 1

a)

v=x(8-2x)(10-2x)

v=x(4x^2-36x+80)

v=4x^3-36x^2+80x

b)

The greatest volume will occur when the velocity of v(x) is zero.

dv/dx=12x^2-72x+80

dv/dx=0 when 

x=(72±√1344)/24 and since 0<x<4  (because of the constraints on x for volume to have meaning)

x≈1.5in (to nearest tenth)

If you were to graph this parabola you would see it have a local maximum peak at about x=1.5 inches.  (it would increase without bound outside the possible values of x for the question, but that is extraneous)